Quantum

As the nineteenth century drew to a close, many physicists felt that the fundamental laws of physics were well understood, and would be a permanent part of humanity's worldview. Although new experimental results, such as the recent discoveries of X-rays, radioactivity, and the electron, could modify the view of the microworld, it was felt that classical mechanics, electromagnetism, thermodynamics, and statistical mechanics, based on classical ideas, would sooner or later provide a satisfactory account of the new phenomena. When William Thomson (Lord Kelvin; 1824–1907) gave a talk at the Royal Institution in London in April 1900 concerning two clouds over the theories of heat and light, he referred not to the recent discoveries but to problems with the ether and the failure of the classical theorem that heat energy should be distributed uniformly among the various possible motions of molecules.

Einstein's relativity theory disposed of the ether (at least for the next century), while the more obscure problems of molecular motion would find their solution in a new mechanics that was born in 1900 from the revolutionary reasoning of a conservative middle-aged physicist, Max Planck. In 1860 Gustav Robert Kirchhoff (1824–1887) had shown that the relative amounts of energy emitted at different wavelengths from any surface was the same for any material and dependent only on the temperature. He also argued that the ratio of the emissive power to the absorptive power of any given material was constant. Kirchhoff conceived of an ideal blackbody that absorbed all the radiation that fell upon it, and he saw that this situation could be realized by an enclosure at constant temperature (such as the interior of a furnace). The radiation from a small hole in such an isothermal enclosure would not depend upon the size or shape of the enclosure or upon the material of the walls. Therefore it had a universal character that called for a fundamental explanation, one that was independent of theories regarding the structure of matter (which were essentially unknown). The spectrum, and hence the color, of the radiation depends only upon the temperature, whether the source be a furnace, the filament of an incandescent lamp, or a glowing star in the sky.

Planck found that explaining the spectrum of blackbody radiation was especially challenging, and hoped to understand its intensity as a function of wavelength or frequency (the two being related by λν = c, c being the velocity of light in vacuum). Born in 1858 in the Baltic city of Kiel to an academic family, he studied in Munich and Berlin, where one of his professors was Kirchhoff, to whose chair he succeeded in 1887 after receiving his Ph.D. in Munich in 1879. Planck lived a long and productive life, undergoing terrible personal losses in both world wars and also during the Nazi period. He died in 1947, after suffering the execution of his younger son for participating in a plot to kill Adolf Hitler.

Planck's Ph.D. thesis dealt with the second law of thermodynamics, which says that heat does not flow from a cooler body to a hotter body (one of several formulations). The first law of thermodynamics expresses the conservation of energy. Planck was attracted to the laws of thermodynamics because they were held to have a universal significance. From about 1897 on he devoted himself to understand the laws of black-body radiation from the standpoint of thermodynamics. Another motivation for him was that experimental scientists at the Physikalisch-Technische Reichsanstalt (the German Bureau of Standards) in Berlin were making an accurate determination of the form of the blackbody spectrum. Using reasoning based first on thermodynamics and then on the statistical mechanics of the Austrian physicist Ludwig Boltzmann (1844–1906), Planck derived a formula that gave precise agreement with the latest experimental results, although even its author, who spent more than a decade afterward trying to improve it, later questioned its original derivation.

Planck's formula for u, the energy per unit volume of radiation in a cavity at frequency, reads:
u_ν = (8πhν³ / c³) (1 / exp(hν / kT) − 1)
where T is the absolute temperature and k is called Boltzmann's constant. Plotted against, the curve for u rises from zero at 0 to form a curved peak with its maximum near h 3 kT, and then falls exponentially to zero for large, where the "1" in the denominator becomes negligible. The peak position that determines the color of the radiation thus increases linearly with the absolute temperature.

The important things to notice about Planck's formula, which he published in 1900, are that it gives the correct spectrum of blackbody radiation and that, aside from the variable quantities frequency and temperature, it contains only the fundamental physical constants h, c, and k. Boltzmann's constant k, which was first introduced by Planck, is the so-called gas constant per mole divided by the number of molecules in a mole (Avogadro's number) and its significance is that kT is a molecular energy. (A monatomic gas molecule has energy (3/2) kT.) The universal constant c is the velocity of light in vacuum. Planck's constant is h 6.626 × 10³⁴ Joule-seconds. Its dimension of energy times time is called "action," and h is called the quantum of action. Evidently Planck's h is a very small quantity and, like k, its most important role is in atomic and molecular physics. The laws of motion of baseballs and planets do not depend upon h, but without it one cannot understand anything about the microworld; its size sets the scale for the elementary particles of which the world is built.